Optimal. Leaf size=487 \[ -\frac {B^2 (b c-a d)^3 g^2 i n^2 x}{3 b d^2}+\frac {B^2 (b c-a d)^2 g^2 i n^2 (c+d x)^2}{12 d^3}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {B (b c-a d)^3 g^2 i n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d^2}+\frac {B (b c-a d)^4 g^2 i n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{12 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \log (c+d x)}{6 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.40, antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2561, 2383,
2381, 2384, 2354, 2438, 2373, 45} \begin {gather*} \frac {B^2 g^2 i n^2 (b c-a d)^4 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3}+\frac {B g^2 i n (b c-a d)^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+3 B n\right )}{12 b^2 d^3}+\frac {B g^2 i n (a+b x) (b c-a d)^3 \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+B n\right )}{12 b^2 d^2}-\frac {B g^2 i n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{12 b^2 d}+\frac {g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{12 b^2}-\frac {B g^2 i n (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b}+\frac {B^2 g^2 i n^2 (b c-a d)^4 \log (c+d x)}{6 b^2 d^3}+\frac {B^2 g^2 i n^2 (c+d x)^2 (b c-a d)^2}{12 d^3}-\frac {B^2 g^2 i n^2 x (b c-a d)^3}{3 b d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2354
Rule 2373
Rule 2381
Rule 2383
Rule 2384
Rule 2438
Rule 2561
Rubi steps
\begin {align*} \int (160 c+160 d x) (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac {160 (b c-a d) (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b}+\frac {160 d (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}\right ) \, dx\\ &=\frac {(160 (b c-a d)) \int (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b}+\frac {(160 d) \int (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b g}\\ &=\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {(80 B d n) \int \frac {(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^2 g^2}-\frac {(320 B (b c-a d) n) \int \frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{3 b^2 g}\\ &=\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {\left (80 B d (b c-a d) g^2 n\right ) \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^2}-\frac {\left (320 B (b c-a d)^2 g^2 n\right ) \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{3 b^2}\\ &=\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {\left (80 B d (b c-a d) g^2 n\right ) \int \left (\frac {b (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {b (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {(-b c+a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b^2}-\frac {\left (320 B (b c-a d)^2 g^2 n\right ) \int \left (-\frac {b (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {b (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {(-b c+a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2 (c+d x)}\right ) \, dx}{3 b^2}\\ &=\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {\left (80 B (b c-a d) g^2 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b}+\frac {\left (80 B (b c-a d)^2 g^2 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}-\frac {\left (320 B (b c-a d)^2 g^2 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d}-\frac {\left (80 B (b c-a d)^3 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d^2}+\frac {\left (320 B (b c-a d)^3 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d^2}+\frac {\left (80 B (b c-a d)^4 g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 d^2}-\frac {\left (320 B (b c-a d)^4 g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {\left (80 B^2 (b c-a d)^3 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b d^2}+\frac {\left (320 B^2 (b c-a d)^3 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{3 b d^2}+\frac {\left (80 B^2 (b c-a d) g^2 n^2\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{3 b^2}-\frac {\left (40 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{b^2 d}+\frac {\left (160 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{3 b^2 d}-\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 d^3}+\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^2 d^3}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b^2}-\frac {\left (40 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {a+b x}{c+d x} \, dx}{b^2 d}+\frac {\left (160 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {a+b x}{c+d x} \, dx}{3 b^2 d}-\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{b^2 d^3}+\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^2 d^2}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {80 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^2 g^2 n^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^2}-\frac {\left (40 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{b^2 d}+\frac {\left (160 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{3 b^2 d}-\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b d^3}+\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 d^2}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac {40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac {40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac {80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 d^3}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 d^2}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac {40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac {40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac {80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log ^2(c+d x)}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 d^3}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^2 d^3}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac {40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac {40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac {80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log ^2(c+d x)}{3 b^2 d^3}+\frac {80 B^2 (b c-a d)^4 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 716, normalized size = 1.47 \begin {gather*} \frac {g^2 i \left (4 (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+3 d (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {4 B (b c-a d)^2 n \left (2 A b d (b c-a d) x+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)-2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}-\frac {B (b c-a d) n \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 n (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}\right )}{12 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{2} \left (d i x +c i \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 2425 vs. \(2 (462) = 924\).
time = 0.83, size = 2425, normalized size = 4.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} g^{2} i \left (\int A^{2} a^{2} c\, dx + \int A^{2} a^{2} d x\, dx + \int A^{2} b^{2} c x^{2}\, dx + \int A^{2} b^{2} d x^{3}\, dx + \int B^{2} a^{2} c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B a^{2} c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A^{2} a b c x\, dx + \int 2 A^{2} a b d x^{2}\, dx + \int B^{2} a^{2} d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int B^{2} b^{2} c x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int B^{2} b^{2} d x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B a^{2} d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A B b^{2} c x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A B b^{2} d x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 B^{2} a b c x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 B^{2} a b d x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 4 A B a b c x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 4 A B a b d x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a\,g+b\,g\,x\right )}^2\,\left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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